Oscillation on Subset

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Theorem

Let $f: D \to \R$ be a real function where $D \subseteq \R$.

Let $x$ be a point in $D$.

Let $S_x$ be a set of real sets that contain (as an element) $x$.

Let $\omega_f \left({I}\right)$ be the oscillation of $f$ on a set $I$ in $S_x$:

$\omega_f \left({I}\right) = \sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$


Let $I \in S_x$.

Let $\omega_f \left({I}\right) \in \R$.

Let $J \in S_x$ be a subset of $I$.


Then:

$\omega_f \left({J}\right) \in \R$
$\omega_f \left({J}\right) \le \omega_f \left({I}\right)$


Proof

Let:

$I, J \in S_x$
$J \subset I$
$\omega_f \left({I}\right) \in \R$

where:

$\omega_f \left({I}\right) = \sup \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$

We need to prove:

$\omega_f \left({J}\right) \in \R$
$\omega_f \left({J}\right) \le \omega_f \left({I}\right)$


We intend to prove that $\omega_f \left({J}\right) \in \R$.

We start by proving that $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ is bounded above and non-empty.


We have that $J$ is a subset of $I$.

Therefore:

$\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ is a subset of $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$

The statement $\omega_f \left({I}\right) \in \R$ means that $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ admits a supremum.

Therefore, $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ is bounded above.

Accordingly:

$\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ is bounded above as $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ is a subset of $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$


We observe that $\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert = 0$ for $y = z = x$.

Therefore, $0 \in \left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ as $x \in J \cap D$.

Accordingly:

$\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ is non-empty


We have that $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ is a real set as $\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert \in \R$ for every $y, z \in D$.

We have shown that $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ is non-empty and bounded above.

Therefore, $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ admits a supremum by the Continuum Property.

In other words:

\(\displaystyle \sup \left\{ {\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}\) \(\in\) \(\displaystyle \R\)
\(\displaystyle \iff \ \ \) \(\displaystyle \omega_f \left({J}\right)\) \(\in\) \(\displaystyle \R\) definition


We finished proving that $\omega_f \left({J}\right) \in \R$.

It remains to prove that $\omega_f \left({J}\right) \le \omega_f \left({I}\right)$.


We have:

$\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ is a subset of $\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$
$\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}$ admits a supremum
$\left\{{\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}$ admits a supremum

Then:

\(\displaystyle \sup \left\{ {\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in J \cap D}\right\}\) \(\le\) \(\displaystyle \sup \left\{ {\left\vert{f \left({y}\right) - f \left({z}\right)}\right\vert: y, z \in I \cap D}\right\}\) Supremum of Subset
\(\displaystyle \iff \ \ \) \(\displaystyle \omega_f \left({J}\right)\) \(\le\) \(\displaystyle \omega_f \left({I}\right)\) definitions

$\blacksquare$